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Full text of "Treatise On Analysis Vol-Ii"

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(1)  The properties (3.8.11) and (3.8.12) are not true for arbitrary topologi-
cal spaces, as is shown by the example of the space E = {a, b} with the
topology (0, {a}, E}. The existence of a distance is essential to the proof of
these two propositions, although it does not feature in the enunciation.

(2)  We may define as in (3.9) the notion of a basis of open sets (or of the
topology) of a topological space E. The criterion (3.9.3) is valid in general. We
can also prove as in (3.9.4) that if there exists a denumerdble basis for the
topology of a topological space E, then there exists an at most denumerable
set which is dense in E; but the converse is not valid for arbitrary topological
spaces (Section 12.4, Problem 6).

By definition, a basis 23 of the topology of a topological space E has the
following property:

The intersection of any two sets 0/23 is a union of sets of 58.

Conversely, let 23 be a set of subsets of a set E. If 23 has the above property
and if E e 23, then the set C of (arbitrary) unions of sets of 93 is a topology for
which 23 is a basis. For it is immediately seen that O satisfies (Oi) and (On).

If E is a topological space and if F is any subset of E, then the set of inter-
sections U n F, where U runs through the set of open subsets of E, satisfies
the axioms (O^ and (On) and is therefore a topology on F. This topology on F
is said to be induced by the topology on E. The set F, endowed with this
topology, is called a subspace of E. With these definitions, all the propositions
of (3.10) are valid for arbitrary topological spaces, with the exception of
(3.10.9), which has to be stated in the following form: if the topology of E has
a denumerable basis, then so does the topology induced on any subset of E.

We have already remarked that the properties of compactness and local
compactness for a metric space depend only on the topology, and not on the
distance which defines the topology. We may therefore speak unambiguously
of compact and locally compact metrizable spaces.

We observe also that all the definitions and results of (3.19) relating to
connectedness are valid for arbitrary topological spaces.

(12.2.1)   Let Cj_, O2 be two topologies on a set E. Then the following properties
are equivalent:

(a)    D2 is finer than O1;

(b)    if EI denotes the topological space obtained by giving E the topology
>!-(/= 1, 2), then the identity mapping ofE2 onto E is continuous;

(c)   for all x e E, every neighborhood ofx in the topology >! is a neighbor-
hood of x in O2.